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| Universitätsbibliothek Graz MS 1, f. 409v (1481) |
"A mouse is at the top of a tree 60 braccia* high, and a cat is on the ground at its foot. The mouse descends 1/2 of a braccio a day and at night it turns back 1/6 of a braccio. The cat climbs one braccio a day and goes back 1/4 of a braccio each night. The tree grows 1/4 of a braccio between the cat and the mouse each day and it shrinks 1/8 of a braccio every night. In how many days will the cat reach the mouse?"
Luca Pacioli, Summa de arithmetica geometria proportioni et proportionalita (1494)Do you ever feel like this cat? All you want is to catch the mouse, but for some reason you always seem to take four steps forward and one step back, and the mouse keeps scurrying around incoherently, and maybe you're crazy but you're starting to think that even the height of the tree is inconsistent?
*The Early Modern Italian braccio is the length of an arm, roughly 23 inches. Unless you have weird arms that grow during the day and shrink at night.
Reduce the problem to one object moving, instead of three. During the day the mouse moves down a half and the cat up one, so change that to the mouse moving down 1.5. The mouse then moves up 1/6, or 2/12 while the cat goes down 1/4 or 3/12. Change that to the mouse moving up 5/12 at night.
ReplyDeleteSince the tree grows 1/4 they get further apart, so change the daily distance change from 1.5 to 1.25. Likewise at night they move 1/8 further apart on top of the 5/12 or 7/24 total. The total change over the full day is they move 23/24 closer to each other.
To sum up, the cat and the mouse move 1.25 closer by day and 7/24 further apart at night. This can be dumped into Excel to find the daily distance apart starting at 60 and you get on day 62 they are 0.58333 apart so on day 63 the cat gets the mouse. Or take 60/(23/24) and you get the cat gets the mouse on day 62.6.
If this is correct, do I win the cat, the mouse or a growing and shrinking tree?
I get the same. If d is the number of days:
ReplyDelete( 1/2 - 1/6 ) d + ( 1 – 1/4 ) d = 60 + ( 1/4 – 1/8 ) d
1/3 d + 3/4 d = 60 + 1/8 d
8/24 d + 18/24 d - 3/24 d = 60
23/24d = 60
d = 62.6
Would it really matter if the tree is growing or not though? A trees grows upward from the top so the mouse and cat should never actually be affected by the top of the tree growing upward and then shrinking a little. In that case the growth is localized and can be taken out of the equation. So then you have a cat that takes 90 days to get to the top and a mouse that take 180 days to get to the bottom I assume they would meet at the 40th braccio or 60 days.
ReplyDeleteIt specifies that the tree is growing between the cat and mouse. It is, after all, an enchanted system. That phrasing actually simplifies the problem. If the tree grew equally along its entire length, the cat and mouse would be separated more by the growth when they were farther apart than when they were closer together. With an equal growth pattern on the stem, at the outset they'd be pushed apart 1/8 bracchio by the tree growth, and when they were about 30 brachia apart, proportional growth would then push them apart only 1/16 of a brachio. I also got 62.6 days.
DeleteRats. Proof that you can't escape word problems even by going back in time.
ReplyDeleteOh my gosh, it's a precursor to those two damned trains leaving different stations at different rates...argh!!
ReplyDeletewell, either 62.3 or 63 - since it's more than a flat 62 the actual meeting of cat and mouse would not happen till the 63rd day. Just like buying 3 gallons of paint even if you only really need 2.6
ReplyDeleteAssuming that the story starts at the beginning of the day (rather than the night), the answer is during the 63rd day (and after 62 nights).
ReplyDeleteSince the net effect is approaching during the day and moving away during the night, I assumed this would be a trick question: that the cat would meet the mouse during day number n, but that if you only looked at the net effect of each day and the following night, you would find n+1 (or even a larger number).
To avoid being tricked, I set op the inequality like this (with n the number of days, hence n-1 the number of nights):
60 + n * (-1/2 -1 +1/4) + (n-1) * (+1/6 +1/4 -1/8) >= 0
As a result, I got n = 62 + 7/23 or about 62.3 (like KateM), rather than 62.6 (like in the earlier comments of Warren, EricD, and Vivianne).
It turned out that there was no catch involved. :-( However, setting it up this way does give a small difference. Although this difference is irrelvant for answering this question, it may be relevant if you want to answer additional questions, like how far did the mouse or the cat have to climb up and down. Although I don't speak Italian (let alone the medieval variety), I could guess most of the words with the help of the translation in this blogpost (apparently, gatta=cat, topo=mouse, elbero=tree, etc.). I checked a scanned version of the book online (see bottom of this page and top of the next) and Pacioli does ask additional questions of this sort!
Neverteless, and to my surprise, Pacioli seems to agree with the 62.6 answer: his text mentions 62 (+) 14/23. I think he got confused by his own problem. ;-)